Cortical surface fMRI (cs-fMRI) has recently grown in popularity versus traditional volumetric fMRI. In addition to offering dimension reduction, removal of extraneous tissue types, and improved alignment across subjects, it is also more compatible with Bayesian spatial models. However, as no spatial Bayesian model has been proposed for cs-fMRI data, most analyses continue to employ a massive univariate approach. Here, we propose a spatial Bayesian model for cs-fMRI, which employs sophisticated spatial processes to model latent activation fields. We make several advances compared with existing spatial Bayesian models for volumetric fMRI. First, we use integrated nested Laplacian approximations (INLA), a highly accurate and efficient Bayesian computation technique, rather than variational Bayes (VB). To identify activations, we utilize an excursions set method based on the joint, rather than the marginal, posterior distribution of the latent fields. Finally, we propose the first multi-subject spatial Bayesian modeling approach, addressing a major gap in the literature. The methods are computationally advantageous and are validated through simulations and two task fMRI studies.